Problem 1

Prove that there are infinitely many natural numbers

a

such that

n^4+a

is not prime for every natural number

n

Step 2 of 4: Choose infinitely many candidates

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In plain words

One explicit infinite family is enough; the problem does not ask for all possible values of $a$ .

a=4r^4,\qquad r=2,3,4,\ldots

Detailed analysis

For every integer $r\ge2$ , set $a=4r^4$ . Distinct values of $r$ give distinct natural numbers $a$ , so it remains only to prove that each such $a$ has the required property.